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Computational Refutation of Quantum Superactivation via Bound Entanglement in Low Dimensions

Yaroslav Murai

Published May 14, 2026• 0 citations
Publisher

Abstract

Smith and Yard (2008) famously demonstrated that two quantum channels with zero quantum capacity can possess a strictly positive joint capacity, a phenomenon known as superactivation. Their proof relies on the existence of Positive Partial Transpose (PPT) states possessing a strictly positive Devetak-Winter private key rate ($K_{DW} > 0$), explicitly citing a 4x4 construction by Horodecki et al. We present a two-tiered computational refutation of this phenomenon. First, using a high-precision global optimization framework with exact Stinespring purifications, we compute $K_{DW}$ across the full 512-dimensional PPT manifold for d=4x4 and find a strict upper bound of $K_{DW} \le -0.68$ bits, physically breaking the foundational protocol. Second, to address recent theoretical defenses concerning macro-scale finite-blocklength scaling ($N \ge 17$), we deploy a 30-qubit hybrid machine learning simulator (equivalent to $N=15$ joint channels). By evaluating the joint topology under strict causal limits and with omniscient encoder prescience, we demonstrate a complete failure to achieve positive capacity, establishing that superactivation is physically impossible in practical quantum systems.

DOI
10.5281/zenodo.20189707

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Smith and Yard (2008) famously demonstrated that two quantum channels with zero quantum capacity can possess a strictly positive joint capacity, a phenomenon known as superactivation. Their proof relies on the existence of Positive Partial Transpose (PPT) states possessing a strictly positive Devetak-Winter private key rate (KDW>0K_{DW} > 0KDW​>0), explicitly citing a 4x4 construction by Horodecki et al. We present a two-tiered computational refutation of this phenomenon. First, using a high-precision global optimization framework with exact Stinespring purifications, we compute KDWK_{DW}KDW​ across the full 512-dimensional PPT manifold for d=4x4 and find a strict upper bound of KDW≤−0.68K_{DW} \le -0.68KDW​≤−0.68 bits, physically breaking the foundational protocol. Second, to address recent theoretical defenses concerning macro-scale finite-blocklength scaling (N≥17N \ge 17N≥17), we deploy a 30-qubit hybrid machine learning simulator (equivalent to N=15N=15N=15 joint channels). By evaluating the joint topology under strict causal limits and with omniscient encoder prescience, we demonstrate a complete failure to achieve positive capacity, establishing that superactivation is physically impossible in practical quantum systems.
Superactivation of quantum capacity is the phenomenon whereby two quantum channels, each with zero quantum capacity, can exhibit a strictly positive capacity when used in tandem. In this work, we explore superactivation in the previously unexplored non-asymptotic regime of finitely many channel uses. We give a definition of finite-blocklength superactivation and propose numerical methods that can certify it. Then, focusing on the 50% erasure and positive-partial-transpose channels considered in the original work on superactivation, we show that as few as 17 uses of the joint channel already enable qubit transmission with a fidelity unattainable by any number of uses of either channel alone, demonstrating a strong finite-blocklength form of superactivation and opening the door to experimental demonstration.
Activation of quantum capacity is a surprising phenomenon according to which the quantum capacity of a certain channel may increase by combining it with another channel with zero quantum capacity. Superactivation describes an even more particular occurrence, in which both channels have zero quantum capacity, but their composition has a nonvanishing one. We investigate these effects for all single-mode phase-insensitive Gaussian channels, which include thermal attenuators and amplifiers, assisted by a two-mode positive-partial-transpose channel. Our result shows that activation phenomena are special but not uncommon. We can reveal superactivation in a broad range of thermal attenuator channels, even when the transmissivity is quite low, or the thermal noise is high. This means that we can transmit quantum information reliably through very noisy Gaussian channels with the help of another Gaussian channel, whose quantum capacity is also zero. We further show that no superactivation is possible for entanglement-breaking Gaussian channels in physically relevant circumstances by proving the nonactivation property of the coherent information of bosonic entanglement-breaking channels with finite input energy.
We derive one-shot upper bounds for quantum noisy channel codes. We do so by regarding a channel code as a bipartite operation with an encoder belonging to the sender and a decoder belonging to the receiver, and imposing constraints on the bipartite operation. We investigate the power of codes whose bipartite operation is non-signalling from Alice to Bob, positive-partial transpose (PPT) preserving, or both, and derive a simple semidefinite program for the achievable entanglement fidelity. Using the semidefinite program, we show that the non-signalling-assisted quantum capacity for memoryless channels is equal to the entanglement-assisted capacity. We also relate our PPT-preserving codes and the PPT-preserving entanglement distillation protocols studied by Rains. Applying these results to a concrete example, the 3-dimensional Werner-Holevo channel, we find that codes that are non-signalling and PPT-preserving can be strictly less powerful than codes satisfying either one of the constraints, and therefore provide a tighter bound for unassisted codes. Furthermore, PPT-preserving non-signalling codes can send 1 qubit perfectly over two uses of the channel, which has no quantum capacity. We discuss whether this can be interpreted as a form of superactivation of quantum capacity.
Recently the power of positive partial transpose preserving (PPTp) and no-signalling (NS) codes in quantum communication has been studied. We continue with this line of research and show that the NS/PPTp/NS∩PPTp codes assisted zero-error quantum capacity depends only on the non-commutative bipartite graph of the channel and the one-shot case can be computed efficiently by semidefinite programming (SDP). As an example, the activated PPTp codes assisted zero-error quantum capacity is carefully studied. We then present a general SDP upper bound QΓof quantum capacity and show it is always smaller than or equal to the “Partial transposition bound” introduced by Holevo and Werner, and the inequality could be strict. This upper bound is found to be additive, and thus is an upper bound of the potential PPTp assisted quantum capacity as well. We further demonstrate that QΓis strictly better than several previously known upper bounds for an explicit class of quantum channels. Finally, we show that QΓcan be used to bound the super-activation of quantum capacity.
In this work a new phenomenon called polaractivation is introduced. Polaractivation is based on quantum polar encoding and the result is similar to the superactivation effect — positive capacity can be achieved with zero-capacity quantum channels. However, polaractivation has many advantages over the superactivation: it is limited neither by any preliminary conditions on the quantum channel nor on the maps of other channels involved in the joint channel structure. We prove that the polaractivation works for arbitrary zero-private capacity quantum channels and we demonstrate, that the symmetric private classical capacity of arbitrary zero-private capacity quantum channels is polaractive.
We study entanglement distillation under operations that remain completely positive under partial-transpose conjugation, a.k.a. PPT channels, in the standard quantum Shannon theory regime of asymptotically vanishing but non-zero error. Our main results are: (a) a regularised formula for the PPT distillable entanglement in terms of a measured-relative-entropy-like quantity; and (b) two different single-letter converses, one based on operator quadratic forms and the other on Hirschman's strengthening of the Hadamard three-line theorem. As an immediate application of (b), we show that the PPT distillable entanglement can be strictly smaller than the regularised Rains bound, thereby resolving an open problem in the theory of entanglement manipulation [Regula et al., NJP 21:103017, 2019]. A gap appears already for 3×33\times 33×3 Werner states: at antisymmetric weight 25/2625/2625/26, a certified upper bound of 0.621070.621070.62107 ebits lies below the (regularised) Rains bound, which equals 2526log⁡25−log⁡23≈0.64766\frac{25}{26} \log_2 5 - \log_2 3 \approx 0.647662625​log2​5−log2​3≈0.64766 ebits. On a different note, (a) implies a faithful lower bound on the PPT distillable entanglement in terms of the entanglement negativity, which provides a quantitative counterpart to the qualitatively known fact that any NPT state is PPT distillable [Eggeling et al., PRL 87:257902, 2001].